There's a fascinating power struggle within the Labour party, but it's not the one you think. Not the tug of war over New Labour's corpse following Peter Mandelson's memoirs: not even the various Eds versus various Milibands.

It's eclipsed by the current emotional dramas of national politics, but offers the sharpest and most urgent choice over Labour's direction. It's the unsung battle between Oona King and Ken Livingstone to be Labour's London mayoral candidate.

For non-Londoners, that sounds like provincial politics. But it's about more than who governs 7 million people, many of them at the sharp end of spending cuts. It's a bigger job than some cabinet roles, and arguably more winnable for Labour than a national election.

Victory in 2012 would be a powerful signal that Labour was back: defeating the Tories with the help, perhaps, of disgruntled Liberal Democrat supporters would destabilise what by then remains of the coalition. And it would provide a practical platform to show how Labour might tackle the recovery, instead of carping from the sidelines.

A similar understanding of London as a springboard to Downing Street led David Cameron to focus sharply on the last mayoral elections. He backed a bold (if futile) experiment in finding new candidates and eventually gambled on Boris Johnson, a risky choice but one dramatising what the new Conservatism was about.

The triumph of a rightwing old Etonian in a liberal metropolis was an important stepping stone on Cameron's route to power. That the accident-prone Johnson has largely not frightened the horses since, even if it's sometimes difficult to remember what he has actually done, strengthened Cameron and confirmed the mayoralty's strategic importance.

The difference for Labour in 2010 is that it doesn't have a leader – and won't before the mayoral candidate is selected, in September. So the party must effectively choose alone.

Helpfully, the differences between King and Livingstone are fairly clear.

Asked at the weekend how they'd manage with no money – the pre-condition for whoever runs London – she talked about getting money from other sources, maybe via complex financial instruments involving the private sector, and pruning non-core services. He vowed to fight the government the way he fought Thatcher from the GLC.

Defiance may be what anxious party members want to hear: it's a simple message on the doorstep, and some Lib Dem defectors may like it. But Labour should consider too what the floating suburban voters and over-55s who backed Johnson want to hear.

In many ways, the candidates are well matched. Both can talk human. Neither are undefeated: King lost her Bethnal Green and Bow parliamentary seat to Respect's George Galloway in a contest capitalising on her voting for the Iraq war, while Livingstone lost the mayoralty to Johnson, the man he's expected to face again. Labour must ask itself, however, which loss is more relevant now.

Above all, it must decide whether by 2012 Labour should be the party of furious opposition to cuts (after cuts have probably been made) or one that bows to the austerity agenda, but cuts more imaginatively and sensitively than the coalition.

The leadership contest should be tackling these questions, of course, but is mostly ducking them. Which leaves King v Livingstone as a proxy test of whether Labour understands how to win again. When everyone is done looking themselves up in Mandelson's index, perhaps the party could reflect on that.

She was ditched as an MP in 2005 and told her IVF treatment had failed. But, as she bids to beat Ken Livingstone as Labour's London mayoral candidate, Oona King says that hardship has made her stronger than ever